This “local to global” approach is a high-dimensional phenomenon that does not hold for graphs! In graphs, the local structure does not reveal any information about the global expansion. To illustrate this, the reader may recall the LPS-Ramanujan graphs [1]} which are (p+1)-regular expander graphs with large girth. One can easily get (p+1)-regular graphs with large girth (and hence locally isomorphic to the LPS ones) which are far from being expanders. In contrast, the Garland method shows that local expansion implies global expansion in the high dimensional case.
